We adopt the L1 approximation [1]}, [2]} of Caputo derivative that appears on the left-hand side in ().
For simplicity, we consider uniform time mesh here.
Let \(\triangle t= \frac{T}{N}\)  be the time step size and \(t_n = n\triangle t, ~0\le n \le N \) .
The L1 approximation is written as
\(\overline{\partial }_t^\alpha u^n \sum _{k = 1}^n b_{n-k}\left(u^k-u^{k-1}\right),\) 
